Identi…cation
Ragnar Nymoen
Department of Economics, UiO
27 February 2009
ECON 4610: Lecture 6
Overview
In this set of notes we address identi…cation in simultaneous
equation models
and in models with measurement errors.
Identi…cation is de…ned and the discussed with the aid of an
example.
We then present general notation for simultaneous equation
models,
and give some general rules for identi…cation is such systems.
Identi…cation with measurement errors at the end.
Main reference is G Ch 13.1;. B Ch 7.1–7.6;K: Ch 10.2. 6N:
D, E and F
ECON 4610: Lecture 6
Identi…cation of econometric models
A parameter in an econometric model is identi…ed if it can be
estimated consistently— not necessarily by OLS though!
Some prefer to express this by saying that an econometric
model is identi…ed if its parameters can be estimated
consistently in an ideal sample.
A main point is that identi…cation is a logical property of the
econometric model, and can be addressed prior to estimation
issues.
ECON 4610: Lecture 6
The simple Keynes model revisited
Let Yt denote GDP in period t D 1, 2, ..., T .
Ct is “endogenous expenditure” and let Xt denote “exogenous
expenditure”.
Assume that Ct depends on GDP, then our example model is
Yt
Ct
D Ct C Xt
(1)
D b1 C b2 Yt C "t , 0 < b2 < 1
(2)
"t is a random disturbance term. We assume that it is white
noise uncorrelated with Xt . For simplicity we assume
normality "t N.0, 2" /.
The parameter of interest is the marginal propensity to
consume b2 .
ECON 4610: Lecture 6
The reduced form of the model
(1) and (2) de…nes a simultaneous equations model. Solution for
the two endogenous variables:
Yt
Ct
11
21
b1
1 b2
b1
D
1 b2
D
D
11
C
D
21
C
12
D
22
D
12 Xt
22 Xt
C
1t
(3)
C
2t
(4)
1
1
b2
b2
1 b2
1
1t
D
1
2t
D
1
ECON 4610: Lecture 6
1
b2
b2
"t
"t
The distribution of Y and C
The Reduced Form written more compactly
Yt
Ct
D
yt
C
1t
(5)
D
ct
C
2t
(6)
where
1t
2
y
N 0,
2t
cy
cy
2
c
j Xt .
The conditional distributions of the stochastic variables 1t and
are binormal with zero expectations and variance matrix:
2
y
cy
cy
2
c
j Xt .
ECON 4610: Lecture 6
(7)
2t
Simultaneity bias
In this example the (main) parameter of interest is the
marginal propensity to consume b2 . In lecture 5 we found that
the OLS estimator bO 2 was inconsistent:
plim bO 2
b2 D
.1 b2 /
Var .Xt /
C1
2
"
The source of this bias is that in the consumption function,
because of the logic of the model, the disturbance "t is
correlated with income Yt .
Idea: What if we obtain OLS estimates from a model where
there is no such correlation, can we derive a consistent
estimator of b2 from that model?
ECON 4610: Lecture 6
Consistency by Indirect least squares (ILS)
Note that from the reduced form we have
22
12
b2
1 b2
D
D b2
1
1 b2
Let OLS estimators of 12 and 22 be denoted with “hats”.
Given the speci…cation of the model it follows that
plim O 22 D 22 and plim O 12 D 12 . Therefore:
plim
O 22
D b2 as well.
O 12
This shows that a there exists a consistent estimator of b2
(and for b1 as well, what is it?). By de…nition, the parameters
of the consumption function are identi…ed.
ECON 4610: Lecture 6
Identi…cation in simultaneous equation models
Looking at the example Keynes model, we see that it is
under-determined: There are more economic variables (3)
than there are equations (2).
Under-determinedness is a necessary condition for
identi…cation in simultaneous equations models.
Why: In the Keynes model: Xt represents observable variation
in Yt that is not due to Ct .
With no Xt in the model, it is determined, and all we can
“infer” from the scatter plot between Ct and Yt is the 45
degree line. With no instrument the model is not identi…ed.
We say that Xt is the instrumental variable, through which
the model becomes identi…ed.
In this case with one degree of freedom, the simultaneous
equations model is exactly identi…ed.
ECON 4610: Lecture 6
Over-identi…cation
What happens if we replace the identity
Yt D Ct C Xt
by
Yt D Ct C Xt C Zt
where Zt is a second exogenous demand component?
Is the model made up of this identity and the consumption
function identi…ed?
Yes, because we can use ILS to obtain two consistent
estimators of b2 .
We say that the model is over-identi…ed because there are
more instrument that is strictly needed for identi…cation.
Over-identi…cation is a luxury, not a problem. And we will
learn methods to use that extra information in an optimal way.
ECON 4610: Lecture 6
Notation for the simultaneous equation model
M endogenous variables .y1 , ...,yM ), and M linear equations,
M structural disturbances (" 1 , ...,"M ).
K exogenous variables (x1 ,.....,xK /.
Let yt , xt denote vectors with observations .t D 1,....,T /, and
let et contain the disturbances.
yt0 0 C x0t B D e0t
(8)
where 0 is a M M coe¢ cient matrix and B is a K M
coe¢ cient matrix. (8) is called the structural form of the
model.
0 1 exists (0 is non-singular). The reduced form:
yt0
D
1
x0t B0
0
0
D xt 5 C vt
C e0t 0
1
5 is the matrix of reduced form coe¢ cients.
ECON 4610: Lecture 6
(9)
The macro model example in matrix notation
yt0
D
Yt
Ct
x0t
D
Xt
1
, 0D
, BD
1
1
1
0
b2
1
0
b1
,
e0t D
0 "t
Multiplying-out according to (8) gives
Yt
Ct
Xt D 0
b2 Yt
Ct
b1 D "t
which of course can be written in the usual way as (1) and (2).
ECON 4610: Lecture 6
A partial market equilibrium model
Let Qt denote the quantity of a good, and Pt the price. A model
of partial market equilibrium is for example
Qt
12 Qt
21 Pt
D
Pt
D
21 x1t
11
22 x1t
12
31 x2t
32 x2t
C "1t , demand(10)
C "2t , supply
(11)
where we have adopted the notation in Greene p 358 eq (13-2),
noting that y1t D Qt and y2t D Pt . Because of the interpretation
as supply and demand schedules we assume 21 > 0 and 12 < 0.
In Greene’s matrix notation the model is (8) with the following
speci…cations:
yt0
D
Qt
x0t
D
1 x1t
Pt
,0D
x21
1
21
12
1
,BD
11
21
31
12
22
32
ECON 4610: Lecture 6
, e0t D
"1t
"2t
Identi…cation in the partial equilibrium model
It useful to proceed in steps. For the simultaneous equation model
(10)-(11) there are 5 cases to consider
1
No exogenous variables
2
One exogenous variable only, in the demand equation:
21 6D 0, and 31 D 22 D 32 D 0.
3
4
5
21
D
31
D
22
D
32
D 0.
One exogenous variable only, in the supply equation:
and 21 D 31 D 22 D 0.
One separate exogenous variable in each equation
32 6D 0 and 31 D 22 D 0
Both exogenous variables enter both equations
ECON 4610: Lecture 6
21
32
6D 0,
6D 0,
Case 1
In this case the only source of variation is the random structural
shocks.
The model will generate a scatter plot like the one below, where
there is no way we can “place” one of both of the curves.
PRICE
QUANTITY
ECON 4610: Lecture 6
Case 2
In this case x1t shifts the demand schedule, but not the supply
curve (rember that x2t is not in the model).
It looks like the supply curve is now identi…ed!
PRICE
QUANTITY
ECON 4610: Lecture 6
Case 3
In this case x2t shifts the supply schedule, but not the demand
curve (x1t is not in the model).
It looks like the demand curve is now identi…ed!
PRICE
QUANTITY
ECON 4610: Lecture 6
Case 4
x2t shifts the supply schedule, but not the demand curve.
x1t shifts the demand curve, not the supply curve.
Qt
12 Qt
D
D
21 Pt
Pt
11
12
21 x1t
32 x2t
C "1t , demand
C "2t ,
supply
(12)
(13)
In this case we cannot readily represent the situation
graphically since both relationships are shifting so the
observation in the P-Q plane will not lie on any particular
“visible” curve.
Graphically we are in much the same situation as in Case 1.
But in this case this tells us nothing about identi…cation.
ECON 4610: Lecture 6
Case 4, cont’d
Intuitively, since x1t and x2t appear in the equations in the
same way as in Case 2 and Case 3, but now jointly,
identi…cation is not lost in Case 4.
We consider a linear combination of (12) and (13), and check
if that relation can (falsely) represent either curve.
Let 0 h 1, then the combined relationship becomes:
.1
h/ C h
12
Qt
D
.1
..1
.1
C.1
h/
h/
h/
11
h
12
21 C h/Pt
21 x1t
h
32 x2t
h/"1t C h"2t
We see that for 0 < h < 1 this equation cannot represent
(12), and it can not represent (13).
For h D 0 we get back (12),
For h D 1 we get back (13),
Hence both are identi…ed.
ECON 4610: Lecture 6
Case 5
The same check as in Case shows that a linear combination of
(12) and (13) is consistent with both of the true structures.
Hence, neither of them are identi…ed.
A second look shows that Case 1 and Case 5 have a common
feature that accounts for the total lack of identi…cation: None
of the equations omits an exogenous variable that is included
in the other equation.
The identi…ed cases are characterized by the omission of one
exogenous variable that appears in the other equation.
This motivates the necessary condition for identi…cation called
the order condition.
ECON 4610: Lecture 6
The order condition
In a simultaneous equation model with M equations,
an equation is identi…ed if the number of exogenous
variables excluded from the equation is equal to or larger
than the number of endogenous variables included in the
equation, less one.
We see that Case 2, 3 and 4 gives exact (or just)
identi…cation according to this Order condition.
In the example with the Keynes model we gave an example of
over-iden…cation: The case where the model was speci…ed
with two separate exogenous demand components in the
general budget equation.
Note than we use this condition (or the equivalent one that
follows) we count any identities (such as the general budget
equation of the Keynes model) as one of the equations of the
model.
ECON 4610: Lecture 6
An equivalent order condition
In a simultaneous equation model with M equations,
an equation is identi…ed if it excludes at least M 1 of
the variables appearing in the model
The see the equivalence, let K i and M i denote the number of
included endogenous and exogenous variables in equation i. The
order condition is then:
K
If we add M
M i on both sides of
K C .M
.M
Ki
i
Mi /
M / C .K
Mi
, and collect terms we get
Ki
.M
i
M
K /
1
Mi / C Mi
1
1
which is the equivalent formulation of the order condition.
ECON 4610: Lecture 6
A su¢ cient (rank) condition
Looking back at Case 4 above, we see that although formally
both equations are identi…ed from the order condition,
suppose that 21 and 32 were zero after all. Then we would
loose identi…cation.: The linear combination cannot be told
apart from the true structural relationships. So we actually
need to assume 21 6D 0 and 32 6D 0.
This generalizes to larger systems where it can happen that
the parameter constellations are exactly such that a linear
combination can become inseparable from one of the
structural equations. To rule that out the following su¢ cient
condition has been formulated:
In a model with M equations, an equation is
identi…ed if and only if it omits at least M 1 variables,
and at least one non-zero (M 1/ (M 1/ determinant
can be formulated from the array of coe¢ cients with
which the omitted variables appear in the other equations
of the model.
ECON 4610: Lecture 6
Linear homogenous restrictions on the parameters
So far we have discussed identi…cation in terms of exclusion
restrictions, which is synonymous with omission of variables.
Exclusion restrictions are special cases of linear restrictions on
parameters. For example 31 D 0 is an exclusion restriction and a
linear restriction. 31 1 D 0 is a linear restriction. 21 C 31 D 0
is another example of a linear restriction.
A generalization of the order conditions is therefore:
In a simultaneous equation model with M equations,
an equation is identi…ed if there are at least M 1 linear
restrictions on the parameters of the equation.
ECON 4610: Lecture 6
Predetermined variables
So far in this set of notes xt has referered to as a vector of
exogenous variables.
However, just as in the regression equation case, the
interpretation of xt can be extended to predetermined
variables.
An explanatory variable is predetermined if it is uncorrelated
with the contemporaneous disturbances in e0t and of all future
disturbances e0t Cj j > 0.
In the same way as in the regression equation case, an
important class of predetermined variables are the lags of the
endogenous variables, ie., yt j , j > 0.
Therefore, the identi…cation conditions above that are
expressed in term of exogenous variables can be re-expressed
in terms of predetermined variables
ECON 4610: Lecture 6
Models with an expectations variable (measurement error)
In lecture 5 we had
yi D
1
C
2 xi
C "i , i D 1, 2, ..., n.
(14)
with all the classical assumptions holding.
xi is an expectations variable that we as econometricians
cannot observe or cannot measure without error.
ui D xi
xi .
(15)
If we try to estimate 2 using the observable (actual) xi and
OLS, that estimator is inconsistent
As we now know that by itself does not mean that 2 is
unidenti…ed. There may be another estimator which is
consistent, and if that is the case, 1 is identi…ed.
ECON 4610: Lecture 6
Identi…cation in the measurement error model
The same substitution as in Lecture 5 gives
yi
xi
D
1
C
2 xi
D xi C ui .
C "i
2 ui ,
(16)
(17)
Regarded as a simultaneous equation model, (16) identi…ed
since it omits one variable in the model, namely xt .
But we also see that this is empty formalism, since the
problem is that xt is unobservable (it is a latent exogenous
variable).
If we introduce the idea of an instrumental variable, zi , and
require of zi that it is correlated with xi but uncorrelated with
"i and ui , we can consider the possibility that the covariances
between yi and zi and between xi and zi give a consistent
estimator.
ECON 4610: Lecture 6
An extended measurement error model
Write the model with a (third) equation that brings zi in as an
exogenous variable:
yi
xi
xi
D
1
D
1
C
2 xi
C "i
C
2 zi
C i,
D xi C ui
2
6D 0
i is a disturbance term with zero mean and which is uncorrelated
with zi . Writing the model in term of the oberveables:
yi
xi
D
1
C
D
1
C
2 xi
2 zi
C "i
C
i
2 ui ,
C ui .
(18)
(19)
Using the order condition we see that the …rst eqation is identi…ed.
If we can …nd a way of utilizing the exogeneity of zi , we will also
…nd that estimator to be consistent.
ECON 4610: Lecture 6
We investigate this by …rst writing (16) in deviation from means
form, and then multiply that equation by zi :
X
.yi
i
.yi
yN /zi
yN /zi
i
2 .x
X
D
D
2
i
x/z
N i C ."i
.xi
"N /z
Xi
x/z
N iC
."
i i
2 .ui
"N /zi
By assumption we have
P
P
plim n1 i .yi yN /zi D Cov .y , z/ 6D 0 plim n1 i .xi
P
P
plim n1 i ." i "N /zi D 0
plim n1 i .ui
so
2
D
2
2
i
.ui
is identi…ed.
ECON 4610: Lecture 6
u/z
N i
x/z
N i D Cov .x, z/ 6D
u/z
N i D0
Cov .y , z/
Cov .x, z/
implying that the Instrumental Variables estimator:
P
yN /zi
i .yi
O IV
2 D P
x/z
N i
i .xi
is consistent, and
u/z
N i
X
(20)
We can also present the argument “back to forth” and start
with suggesting
P
yN /zi
i .yi
O IV
2 D P
x/z
N i
i .xi
on the basis that it uses the exogenous variable zi instead of
xi in the OLS estimator.
IV
We then need to show that plim O 2 D 2 .
P
1 P
x/
N C ."i
.yi yN /zi
IV
2 .xi
i
i
n
O2 D P
D
1 P
x/z
N i
i .xi
i .xi
n
"N /
x/z
N i
By the exogeneity of zi we see that the “plims” in the
previous slide apply, so that
IV
plim O 2 D
2 Cov .xi , zi /
Cov .xi , zi /
D
ECON 4610: Lecture 6
2.
2 .ui
u/
N zi